Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 5x - 16$, and $ m \angle BOC = 5x - 64$, find $m\angle BOC$. $O$ $A$ $C$ $B$
Solution: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {5x - 16} + {5x - 64} = {90}$ Combine like terms: $ 10x - 80 = 90$ Add $80$ to both sides: $ 10x = 170$ Divide both sides by $10$ to find $x$ $ x = 17$ Substitute $17$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 5({17}) - 64$ Simplify: $ {m\angle BOC = 85 - 64}$ So ${m\angle BOC = 21}$.